To determine the number of ways Nina can arrange her five books on the shelf, we need to calculate the number of possible permutations of five distinct items.

The formula for the number of permutations of $n$ items is given by:

$n! = n \times (n-1) \times (n-2) \times \ldots \times 1$

Since Nina has five books, we use $n = 5$:

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

So, Nina can arrange her five books in 120 different ways.

Would you like further details or have any questions?

Here are some related questions:

1. What if Nina had six books? How many arrangements would be possible?
2. How does this change if some books are identical?
3. What is the probability that Nina picks a specific order randomly out of all possible arrangements?
4. How can factorial calculations be simplified using patterns?
5. In how many ways can Nina pick only three books to arrange on her shelf?

Tip: Factorials grow very quickly; even a small increase in the number of items can lead to a large increase in possible arrangements.